Finding a feasible search direction.
With all gradients determined via IPA, we are almost ready to perform a gradient descent method. However, in order for the tracking combination constraint to remain satisfied, the step direction with respect to the tracking constraints, denoted by \(p_\alpha \) , must be chosen appropriately, i.e., its components must add up to zero for each tracking period \(\ell \)  and agent \(j\) . We do so by computing a feasible step direction via the least squares problem

p J - p22
0 i j + pi j1 , i, j,
m=1M pm j = 0 ,j,

where \(p_{\alpha _{\ell i j}}\)  is the \(i\) th component of the step direction for agent \(j\)  in the \(l\) th tracking period. The global solution of this convex optimization problem yields the steepest gradient of \(J\)  respecting the tracking parameter constraints. For the remaining parameters, we directly use the IPA gradients. In order to achieve cost function descent at each iterate, we choose a step length via the Armijo backtracking technique [1]}.
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